Chapter 19 NumericsNumerics in General in Generalwebbuild.knu.ac.kr/~bykang/chapter19.pdf · 2009-11-20 · Chapter 19 NumericsNumerics in General in General Bo-Yeong Kang, 2009 Kyungpook

Chapter 19Chapter 19Chapter 19Chapter 19NumericsNumerics in Generalin General

Bo-Yeong Kang, 2009Kyungpook National University

1

What is What is numerical analysis ?y

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The study of algorithms to sol e the problems thatsolve the problems thatthere is no solution formula

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19.1 Introduction

The steps from problems p pto answer

ModelingModelingChoosing a numeric methodsprogrammingprogrammingDoing the computationInterpreting the results

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19.1 Introduction

Real number

Fl ti i t f f Floating point form of numbersnumbers

0.6247*103, 0.735*10-13, -0.2000*10-1

Number of significant digits is kept fixedg g pDecimal point “floating”

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19.1 Introduction

Floating point form of Floating point form of numbersnumbers

a = m*10n , 0.1≤|m| < 1On computers, m & n is limitedp ,ā = m*10n , m=0.d1d2…dk, d1>0

mantissa

exponent

mantissa

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19.1 Introduction

Floating point form of n mbe snumbers

ā = m*10n , m=0.d1d2…dk, d1>0mantissa

exponent

exponent

tisign mantissaIBM 3000 & 4000 series

Single precision floating point number1 sign, 7 exponent, 24 mantissa, 32 bits

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19.1 Introduction

Floating point form of oat g po t o onumbers

The number outside that range occursUnderflow when the number is smaller (zero)Overflow when it is larger (halt)

exponent

sign mantissa

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19.1 Introduction

Roundoff(끝처리)mantissa

Roundoff(끝처리)

0.d1d2…dkdk+1dk +2dk+3 …Roundoff ruleDiscard (k+1)th and all subsequent decimalsDiscard (k+1)th and all subsequent decimalsIf the number discarded is less than half a unit in kth place, leave kth decimal unchanged(rounding down)If it is greater than half a unit, round add one to the kth decimal If it is greater than half a unit, round add one to the kth decimal (rounding up)If it is exactly half a unit, round off to the nearest even decimal

Rounding up/down happens about equallyg p/ pp q y

ex)3.45-3.4, 3.55-3.6

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19.1 Introduction

Error in RoundingError in Rounding

ā= fl(a), floating point approximation of a by rounding

|m-m| ≤ ½ * 10-k (ex. |0.45-0.4| ≤ ½ * 10-1)

|m| ≥ 0.1 , 1/|m| ≤ 10|(a- ā)/a| ≈|(m-m)/m| ≤ ½ * 101-k

Error bound in rounding

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19.1 Introduction

Errors of Numeric Results

Finding the results – ApproximationRounding errorsExperimental errors from measurementsT ncating e o s f om p emat el b eaking offTruncating errors from prematurely breaking off

Formulas for ErrorsFormulas for Errorsє = a – ãa = ã + є , True value = Approximation + errora ã + є , True value Approximation + errorã = 10.5, a=10.2, є = -0.3

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19.1 Introduction

Errors of Numeric Results

є = |a – ã|(absolute error) or ã – a| |( )єr = є/a = (a-ã)/a = Error/True value (relative error )This look useless because a is unknown

/ã єr = є/ã This still looks problematic because є is unknown

One can obtain in practice is an error bound for ã, |є| ≤ , hence |a-ã| ≤ | | | | |єr|≤ r , hence |(a-ã)/a| ≤ r

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19.1 Introduction

E P tiError Propagation

How errors at the beginning and in later steps propagate into the computation and affect accuracy

(a) Subtractionx = x+є y=ỹ+є |є |≤ |є | ≤ x = x+є1 , y=ỹ+є2 , |є1|≤ 1 , |є2| ≤ 2 є = |x-y – (x-ỹ)|

= |x-x – (y-ỹ)|True value subtraction

Approximation subtraction= |x x (y ỹ)|= | є1 – є2 | ≤ |є1|+|є2| ≤ 1 + 2

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19.1 Introduction

Error PropagationError Propagation

(b) Relative error of multiplicationx = x+ є1 , y=ỹ+ є2 , |єr1|≤ r1 , |єr2| ≤ r2

|( ỹ)/ |єr = |(xy – xỹ)/xy|= |(xy – (x- є1)(y-є2))/xy|

|( )/ |= |(є1y+є2y-є1є2)/xy| ≈ |(є1y+є2x)/xy|

| / | | / | | | | | ≤ | є1/x |+ |є2 /y| = |єr1|+|єr2| ≤ r1 + r2

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19.1 Introduction

Error PropagationError PropagationTh 1 E P iTh 1 E P iTh.1 Error PropagationTh.1 Error Propagation

(a)In addition and subtraction, an error boundf th lt i i b th f thfor the results is given by the sum of the errorbounds for the terms(b)I lti li ti d di i i b d(b)In multiplication and division, an error boundfor the relative error of the results is given byh f h b d f h l i fthe sum of the bounds for the relative error of

the given numbers

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19 2 Solution of 19.2 Solution of Equation by Iterationq o y o

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19.2 Solution of Equation by Iteration

F(x) = 0

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F(x) = 0F(x) 0Zeros of Bessel function ?

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Algebraic equationAlgebraic equationTranscendental equation

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F(x) = 0

No formula for exact solutionApproximation

Iteration method

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Iteration methodIteration method

Initial guess x0step by step x1 x2 step by step x1 x2 …

Fi d i t it tiFixed point iterationNewton’s method

Secant method

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Fixed-point iterationp

Transform f(x) 0 to x g(x)Transform f(x)=0 to x=g(x)Choose x0x1 =g(x0)x2 =g(x1)2 g( 1)…Xn+1 =g(xn)n+1 g n

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Ex1. Fixed-point iterationp

Transform to x=g(x)

Converge

Diverge24



Divide by x

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Indexing Indexing

Extract the index terms Extract the index terms Weights Data structure

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Header fileEx. stdlib.h

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Data typeData type

Int integerInt, integer

Long, long integer

Float, floating-point

Double, double precision floating point

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Indexing Indexing


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Indexing Indexing


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x0 =1 0x0 =1.0

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3 0x0 =3.0

(+,-)1.#INF, infinite numberNaN not a number =NaN, not a number =(+,-)1.#IND (indeterminate)

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lower part, the slope of g1(x) is less than the slope of y=xof y x

upper part, the slope of g1(x) is steeper than the slope of y=x

|g1’(x)| < 1 is sufficient for convergence

1.0 3.0 33

C f Fi d P i t 19.1 Introduction

Convergence of Fixed-Point IterationIteration

Th 1 CTh 1 CTh.1 ConvergenceTh.1 Convergence

Let x=s be a solution of x=g(x) and supposethat g has a continuous derivative in someinterval J containing s. Then if |g’(x)| ≤ K < 1 inJ, the iteration process defined by (3) convergesfor any x0 in J, and the limit of the sequence {xn}is s.

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C f Fi d P i t 19.2 Solution of Equation by Iteration

Convergence of Fixed-Point IterationIteration

|g’(x)|<1

SS

J35

Proof19.2 Solution of Equation by Iteration

Proof

|g’(x)|<1

SS

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Transform to x=g(x)

| ’( )| |2 /3||g1’(x)| = |2x/3|<1-3/2 < x < 3/2 의 어떤 x0에서라도 수렴

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Divide by x

|g1’(x)| = |1/x2|<1 1 1 의 어떤 에서라도 수렴-∞< x <-1, 1<x<∞ 의 어떤 x0에서라도 수렴

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Secant method

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Newton’s Method

F(x)=0

Commonly usedSimplicity and great speedSimplicity and great speed

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Newton’s Method

Idea

Tangent

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Newton’s Method

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Fail, Try another x0

Success

Fail, Try another x0, y 0

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Ex3. Square Rootq

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45

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Ex4. Transcendental Equationq

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Ex5. Algebraic Equationg q

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Order of an Iteration methodSpeed of Convergence

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Subtract g(s)=s

xn+1 – s = -g’(s) єn + 1/2 g’’(s) єn2 + …

є = g’(s) є + 1/2 g’’(s) є 2 + -єn+1 = -g (s) єn + 1/2 g (s) єn2 + …

єn+1 = g’(s) єn - 1/2 g’’(s) єn2 + …

(a) єn+1 ≈ g’(s) єn in the case of first order

(b) є +1 ≈ -1/2 g’’(s) є 2 in the case of second order(b) єn+1 ≈ 1/2 g (s) єn in the case of second order

Ex. єn=10-k, єn+1 = c*(10-k)2=c*10-2k

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C f N t ’ 19.2 Solution of Equation by Iteration

Convergence of Newton’s MethodMethod

єn+1 = g’(s) єn - 1/2 g’’(s) єn2 + …єn+1 g (s) єn 1/2 g (s) єn + …

єn+1 ≈ -1/2 g’’(s) єn2 , second order

if f’ and f’’ are not zero at a solution s

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Taylor Series

미분가능한 어떤 함수를 다항식의 형태미분가능한 어떤 함수를 다항식의 형태로 근사화하는 방법

어떤 한 점에서 그것의 미분값으로 계산되는 항들의 무한 합으로 표현

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Taylor SeriesTaylor Series

f(x) = c0+c1(x-a)+c2(x-a)2+c3(x-a)3+…(단, |x-a|<R 때 수렴)( | | )

x=a, f(a) = c0

f’(x) = c1+2c2(x-a)+3c3(x-a)2+…x=a, f’(a) = c1

f’’(x) = 2c2+3*2c3(x-a)+…x=a, f’’(a) = 2c2, f’’(a)/2 = c2

Cn =

f’’’(x) = 3*2c3+…x=a, f’’’(a) = c3, f’’’(a)/(3*2) = c3…

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Taylor SeriesTaylor SeriesThe Taylor series of a real or complex function ƒ(x) y p ƒ( )that is infinitely differentiable in a neighborhood of a real or complex number a, is the power series

where n! denotes the factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a; the zeroth derivative of ƒ is derivative of ƒ evaluated at the point a; the zeroth derivative of ƒ is defined to be ƒ itself and (x − a)0 and 0! are both defined to be 1.

In the particular case where a 0 the series is also In the particular case where a = 0, the series is also called a Maclaurin series.

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처음 몇 항까지를 선택함으로써 x = a 주변에서의 f(x)의 근사식으로 사용

The exponential function y = ex (continuous red line) and the corresponding Taylor polynomial of degree four (dashed green line) around the originline) around the origin.

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The exponential function (in blue) The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).

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Secant method

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Secant Method

f’(x)

A far more difficult expressionComputationally expensiveComputationally expensive

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Secant Method

초기치 2개 필요초기치 2개 필요

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Ex8. Secant Method

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Convergence of Secant Method

Evaluation of derivativesAvoided

Convergence Almost quadratic like NewtonAlmost quadratic like Newton

|є | ≈ const| є |1 62|єn+1| ≈ const| єn |1.62

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Secant method

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19.3 Interpolation

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19.3 Interpolation

approximate values approximate values of a function f(x) for an x

between different x-values x0,x1,…,xn

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19.3 Interpolation

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19.3 Interpolation

Lagrange Interpolation

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19.3 Interpolation

y-f1=(f1-f0/x1-x0)(x-x1)

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19.3 Interpolation

Lagrange Interpolationg g p

IdeaIdea

(x0,f0), (x1,f1)x=x0 P1(x)=f0x=x0, P1(x)=f0x=x1, P1(x)=f1

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19.3 Interpolation

(x0,f0), (x1, f1)x=x0, P1(x)=f0x=x1 P1(x)=f1x=x1, P1(x)=f1

P (x ) =f L +f L =f (L =1 L =0)P1(x0) =f0L0+f1L1 =f0 (L0=1, L1=0)P1(x1) =f0L0+f1L1 = f1 (L0=0, L1=1)

L0= (x-x1)/(x0-x1)L0= (x x1)/(x0 x1)L1= (x-x0)/(x1-x0)

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19.3 Interpolation

y-f1=(f1-f0/x1-x0)(x-x1)y=(f1-f0/x1-x0)(x-x1)+f1={(f1-f0)(x-x1)+f1(x1-x0)}/(x1-x0)={(-f0)(x-x1)+f1(x-x1)+f1(x1-x0)}/(x1-x0)={(-f0)(x-x1)+f1(x-x0)}/(x1-x0) =(-f0)(x-x1) /(x1-x0)+f1(x-x0) /(x1-x0)= f0(x-x1) /(x0-x1)+f1(x-x0) /(x1-x0)= f0L0+f1L1 74

19.3 Interpolation

Ex1. Linear Lagrange Interpolation

P1(x) =f0L0+f1L1 ,L = (x x )/(x x ) L = (x x )/(x x ) L0= (x-x1)/(x0-x1), L1= (x-x0)/(x1-x0)

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19.3 Interpolation

L0= (x-x1)(x-x2)/(x0-x1)(x0-x2)L0 (x x1)(x x2)/(x0 x1)(x0 x2)L1= (x-x0)(x-x2)/(x1-x0)(x0-x2)L2= (x-x0)(x-x1)/(x2-x0)(x2-x1)

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19.3 Interpolation

Quadratic InterpolationQ p

three pointsthree points(x0,f0), (x1,f1), (x2,f2)

x=x0, P2(x)=f0x=x1 P2(x)=f1 x=x1, P2(x)=f1 x=x2, P2(x)=f2

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19.3 Interpolation

P2(x) =f0L0+f1L1+f2L2

(x0,f0), (x1,f1), (x2,f2)x=x (L =1 L =0 L =0) x=x0, (L0=1, L1=0, L2=0) x=x1, (L0=0, L1=1, L2=0) x=x2, (L0=0, L1=0, L2=1)

L0= (x-x1)(x-x2)/(x0-x1)(x0-x2)L ( )( )/( )( )L1= (x-x0)(x-x2)/(x1-x0)(x0-x2)L2= (x-x0)(x-x1)/(x2-x0)(x2-x1)2 ( 0)( 1)/( 2 0)( 2 1)

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19.3 Interpolation

Ex2. Quadratic Lagrange Interpolation

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19.3 Interpolation

L ( )( )/( )( )L0= (x-x1)(x-x2)/(x0-x1)(x0-x2)L1= (x-x0)(x-x2)/(x1-x0)(x0-x2)L2= (x-x0)(x-x1)/(x2-x0)(x2-x1)

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19.3 Interpolation

L = (x x )(x x )/(x x )(x x )L0= (x-x1)(x-x2)/(x0-x1)(x0-x2)L1= (x-x0)(x-x2)/(x1-x0)(x0-x2)Code 작성Code 작성

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19.3 Interpolation

General Lagrange I t l ti P l i lInterpolation Polynomial

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Error Estimate

83

19.3 Interpolation

Ex3. Error Estimate of Linear Interpolation

9 0 ≤ t ≤ 9 59.0 ≤ t ≤ 9.5

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19.3 Interpolation


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19.3 Interpolation


86

InterpolationInterpolation

Linear Lagrange InterpolationQuadratic Lagrange Interpolation Quadratic Lagrange Interpolation General Lagrange Interpolation

Error Estimate

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19 5 Numeric Integration 19.5 Numeric Integration & Differentiation

88

N i l ti f Numeric evaluation of integralsintegrals

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Rectangular Rule

middle point of 1-st subinterval

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Trapezoidal Rule

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19.5 Numeric Integration

Ex1. Trapezoidal Rule

F(0)+f(1)

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19.3 Interpolation

0.746824

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19.3 Interpolation

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19.3 Interpolation

0.746824

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Error Bounds and Estimate for Trapezoidal Rule

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98


99


f

100


Ex2. Error Estimation for Trapezoidal Rule

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Simpson’s Rule Simpson s Rule of Integration

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Piecewise constant approximation ppRectangular rule

Piecewise linear approximationTrapezoidal ruleTrapezoidal rule

Piecewise quadratic approximationSimpson’s rulep

103

Divide the intervali t binto even number

of equal subintervalsof equal subintervals

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105

-h -2h h -h 2h h

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107

108

Error of Simpson’s Rule Error Bound


Error of Simpson s Rule, Error Bound, Degree of Precision

Because Simpson rule uses second order, if n=2, error ofSimpson rule has f(3) derivatives ((5) in pp.800 & (3*) in pp.819).B hi th bl i T l l i lBy approaching the problem using Taylor polynomial, we canderive the error involves the fourth derivative of f(4) .Simpson’s rule gives exact results when applied to anypolynomial of degree three or lesspolynomial of degree three or less.

109




110




111




• Newton–Cotes formulas are a group of formulas for numerical integration based on evaluating the integrand at equally-spaced points

Closed Newton-cotes formula, [a,b]Open Newton-cotes formula, (a,b)

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Numeric Differentiation

113

Computation of values of derivative of f(x)

from a given value of f(x)from a given value of f(x)

114

Numeric differentiation should be avoided whenever possiblewhenever possible

115

큰 값의 작은 차를큰 값의 작은 차를작은 양으로 나눈 것의 극한

약간의 부정확한 f(x) 값이라도약간의 부정확한 f(x) 값이라도f’(x)에 큰 영향을 줄 수 있음

116

19.5 Numeric Differentiation

f(x )f(x1)f(x0)

x0 x1x1/2

h

117


f(x) = x4

f’(1.5) = ?

f(x )f(x1)f(x0)

1.5x1/2

2.0x1

1.0x0

118

x1/2

h=1

x1 x0

f’’(x ) ?


f’’(x1) = ?

119

19.3 Interpolation

f(x) x4f(x) = x4

f’’(2) = ?

120

Lagrange polynomial Lagrange polynomial approximation, differentiation

P2(x) =f0L0+f1L1+f2L2-h -2h

L0= (x-x1)(x-x2)/(x0-x1)(x0-x2)L1= (x-x0)(x-x2)/(x1-x0)(x0-x2)L ( )( )/( )( )L2= (x-x0)(x-x1)/(x2-x0)(x2-x1)

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Lagrange polynomial Lagrange polynomial approximation, differentiation

x=x0

x=x1

x=x2122

Numeric Integration gRectangular RuleTrapezoidal RuleTrapezoidal Rule

Simpson RuleError Estimate

DifferentiationDifferentiation

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Chapter 19 NumericsNumerics in General in Generalwebbuild.knu.ac.kr/~bykang/chapter19.pdf · 2009-11-20 · Chapter 19 NumericsNumerics in General in General Bo-Yeong Kang, 2009 Kyungpook